Chapter 2 Review of Random Variables

Updated: April 25, 2021

This chapter reviews the probability concepts that are necessary for the modeling and statistical analysis of financial data presented in this book. This material is typically covered in an introductory probability and statistics course using calculus. In the course of the review, many examples related to finance will be presented and some important risk concepts, such as value-at-risk, will be introduced. The examples will also show how to use R for probability calculations and working with probability distributions.

The chapter is outlined as follows. Section 2.1 reviews univariate random variables and properties of univariate probability distributions. Important univariate distributions, such as the normal distribution, that are used throughout the book for modeling financial data, are described in detail. Section 2.2 covers bivariate probability distributions for two random variables. The concepts of dependence and independence between two random variables are discussed. The measures of linear dependence between two random variables, covariance and correlation, are defined and the bivariate normal distribution is introduced. Properties of linear combinations of two random variables are given and illustrated using an example to describe the return and risk properties of a portfolio of two assets. More than two random variables and Multivariate distributions are briefly discussed in section 2.3.

The R packages used in this chapter are mvtnorm and sn. Make sure these packages are downloaded and installed prior to replicating the R examples in this chapter.